Explanation Given: The magnitude of component vector force F , F 1 = 80 lb . The magnitude of component vector force F , F 2 = 130 lb . The angle between F 1 and F ′ is 30 ° The angle between F ′ and x axis is 40 ° Illustrate the magnitude of components force F 1 and F 2 in x , y , and z directions as shown below in Figure (1). Illustrate the direction angles with the magnitude of component force F 1 as shown below in Figure (2). Illustrate the direction angles with magnitude of component force F 2 as shown below in Figure (3). Use trigonometry to calculate the magnitude of the components from Figure (1). F z = ( F 1 sin 30 ° ) lb (I) F ′ = ( F 1 cos 30 ° ) lb (II) F x = ( F ′ cos 40 ° ) lb (III) F y = ( F ′ sin 40 ° ) lb (IV) Here, the component force in x , y , and z axes is F x , F y , and F z respectively, the sum of forces, F x and F y is F ′ . Calculate the resultant vector component force F 1 in terms of unit vectors ( i , j , and k ) that represents the direction cosines of F . F 1 = F x i + F y ( − j ) + F z k = F 1 cos α i − F 1 cos β j + F 1 cos γ k (V) Conclusion: Substitute 80 lb for F 1 in Equation (I). F z = ( 80 sin 30 ° ) lb = 80 × 0.5 = 40 lb Substitute 80 lb for F 1 in Equation (II). F ′ = ( 80 cos 30 ° ) lb = 80 × 0.866 = 69.28 lb Substitute 69.28 lb for F ′ in Equation (III). F x = ( 69.28 cos 40 ° ) lb = 69.28 × 0.766 = 53.06 lb Substitute 69.28 lb for F ′ in Equation (IV). F y = ( 69.28 sin 40 ° ) lb = 69.28 × 0.642 = 44.47 lb Substitute 53.06 lb for F x , 44.47 lb for F y , and 40 lb for F z in Equation (V). F 1 = [ 53.06 i − 44.47 j + 40 k ] (VI) Thus, the resultant vector component force F 1 is [ 53.06 i − 44.47 j + 40 k ] _ . Compare the Equations (V) and (VI). F 1 cos α 1 = 53.06 F 1 cos β 1 = 44.47 F 1 cos γ 1 = 40 Calculate the direction angle α 1 by substituting 80 lb for F 1 . α 1 = cos − 1 ( 53

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 2.6, Problem 64P

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The vector force F1 and F2 as a Cartesian vector.

The coordinate direction angles of the vector force F1 (α1,β1,γ1) and F2 (α2,β2,γ2).

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Chapter 2.6, Problem 63P

Chapter 2.6, Problem 65P

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Chapter 2 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

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The vector force F 1 and F 2 as a Cartesian vector. The coordinate direction angles of the vector force F 1 ( α 1 , β 1 , γ 1 ) and F 2 ( α 2 , β 2 , γ 2 ) .

Solution Summary: The author calculates the vector force F_1 and f2, as a Cartesian vector, and the coordinate direction angles of the force.

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The vector force F1 and F2 as a Cartesian vector.

The coordinate direction angles of the vector force F1 (α1,β1,γ1) and F2 (α2,β2,γ2).

Want to see the full answer?

Chapter 2 Solutions

The vector force F 1 and F 2 as a Cartesian vector. The coordinate direction angles of the vector force F 1 ( α 1 , β 1 , γ 1 ) and F 2 ( α 2 , β 2 , γ 2 ) .

Solution Summary: The author calculates the vector force F_1 and f­2, as a Cartesian vector, and the coordinate direction angles of the force.

Videos

The vector force F1 and F2 as a Cartesian vector. The coordinate direction angles of the vector force F1 (α1,β1,γ1) and F2 (α2,β2,γ2).

Want to see the full answer?

Chapter 2 Solutions

Solution Summary: The author calculates the vector force F_1 and f2, as a Cartesian vector, and the coordinate direction angles of the force.

The vector force F1 and F2 as a Cartesian vector.

The coordinate direction angles of the vector force F1 (α1,β1,γ1) and F2 (α2,β2,γ2).